In Bayesian inference a family of densities \(\mathbb {P}_\Omega =\lbrace  p(\cdot |\varphi ,\theta ): (\varphi ,\theta ) \in \Omega \rbrace \)  with parameter space \(\Omega \)  is specified for unknown parameters \(\theta ,\varphi \)  and belief about the true parameters \((\theta ^*,\varphi ^*)\)  is updated by the observed data using Bayes' rule. If the model is well specified \(p^*\in \mathbb {P}_\Omega \) , then under regularity conditions, the posterior concentrates at the true parameter values as the number of observations increases. If the parametric model is misspecified \(p^*\notin \mathbb {P}_\Omega \)  then, under regularity conditions, the posterior concentrates at the pseudo-true parameter values minimizing the Kullback-Leibler divergence between \(p^*\)  and \(p(\cdot |\varphi ,\theta )\)  [1]}.
In these settings the Maximum Likelihood Estimator (MLE) is a natural estimator for the parameters minimising the Kullback-Leibler divergence [2]}. The pseudo-truth is given by the limiting MLE taken on large data. The asymptotic behaviour of the Bayes posterior distribution for misspecified parametric models is considered in [3]}. A covariance matrix guaranteeing the correct asymptotic Frequentist coverage of the pseudo-true parameters was given by [4]}.
